Your description is a bit oversimplified, but basically yes.

They send in a delegation that includes diplomats and experts in (usually in cars like this whatever is to be investigated and then they "do their thing". Also, afaik the UN only sends people when their safety is guaranteed by whoever is holding political/military power in the given area.

In case of the chemical attacks, they'll have chemical experts that take probes of the ground, talk to doctors and maybe see patients, might take their own tissue samples from bodies of people killed in the attack, might interview people who (claim to) have witnessed the attack, and so forth.

Another example of UN investigations would be in Iraq (before the US attacked) where the UN looked for WMDs. IIRC the chief inspector was Hans Blix of Sweden (so no, they don't have to be Swiss) the former head of the UN International Atomic Energy Agency. Here's some info on that mission